I own a '97 ES300 and have the same problem. The engine STARTS and then stalls. This happens about 10% of the time. I can tell if the engine is going to stall by paying attention to the tachometer right after turning the key to start the engine. Normally, when I start a cold engine, the engine speed starts off at around 1,500 RPM. However, SOMETIMES, the engine STARTS with engine speed less than 300 RPM. I have to tap on the gas pedal so that the speed reaches about 3,500 RPM for about a second. After I do this, the engine does not stall.
If it were a battery problem, it would happen 100% of the time. If it were a problem with either the fuel filter or the fuel system, the engine would probably stall while driving or while stopping at a red light. However, as brianchough wrote, once the car gets going, there is no problem.
